On the other hand, the idea of using the Morse index of a solution for a semilinear elliptic equation was first explored by Bahri and Lions [1]} to get further proved that when \(1<p<\frac{N+2}{N-2}\) , no sign-changing solution exists for (REF ). To prove this result, they first deduced some integrable conditions on the solution based on finite Morse index; then they used the Pohozaev identity to prove the nonexistence result.
So, motivated by [2]}, they used blow-up argument to obtain a relevant \(L^\infty \) -bound for solutions of semilinear boundary value problems in bounded domain from the boundedness of Morse index (see also [1]}, [4]}, [5]}, [6]}, [7]}). We mention also that when the Palais-Smale; or the Cerami compactness conditions for the energy functional do not seem to follow readily, the proof of existence of solutions is essentially reduced to deriving \(L^\infty \) -estimate from Liouville-type theorems via Morse index (see for instance [8]}, [9]}, [10]}). After these works, many authors investigated various Liouville type theorems for solutions with finite Morse indices in subcritical case such as problems with Neumann boundary condition, Dirichlet-Neumann mixed boundary and nonlinear boundary conditions (see [11]}, [12]}, [5]}, [14]}, [15]}, [16]}, [17]}). In the supercritical case, the finite Morse index solutions to the corresponding nonlinear problem (REF ) have been completely classified by Farina [4]}.
